3.128 \(\int x^m \coth ^{-1}(\tanh (a+b x)) \, dx\)

Optimal. Leaf size=37 \[ \frac{x^{m+1} \coth ^{-1}(\tanh (a+b x))}{m+1}-\frac{b x^{m+2}}{m^2+3 m+2} \]

[Out]

-((b*x^(2 + m))/(2 + 3*m + m^2)) + (x^(1 + m)*ArcCoth[Tanh[a + b*x]])/(1 + m)

________________________________________________________________________________________

Rubi [A]  time = 0.027411, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2168, 30} \[ \frac{x^{m+1} \coth ^{-1}(\tanh (a+b x))}{m+1}-\frac{b x^{m+2}}{m^2+3 m+2} \]

Antiderivative was successfully verified.

[In]

Int[x^m*ArcCoth[Tanh[a + b*x]],x]

[Out]

-((b*x^(2 + m))/(2 + 3*m + m^2)) + (x^(1 + m)*ArcCoth[Tanh[a + b*x]])/(1 + m)

Rule 2168

Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^(m + 1)*v^
n)/(a*(m + 1)), x] - Dist[(b*n)/(a*(m + 1)), Int[u^(m + 1)*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m
, n}, x] && PiecewiseLinearQ[u, v, x] && NeQ[m, -1] && ((LtQ[m, -1] && GtQ[n, 0] &&  !(ILtQ[m + n, -2] && (Fra
ctionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ[n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] &&  !IntegerQ[m]
) || (ILtQ[m, 0] &&  !IntegerQ[n]))

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^m \coth ^{-1}(\tanh (a+b x)) \, dx &=\frac{x^{1+m} \coth ^{-1}(\tanh (a+b x))}{1+m}-\frac{b \int x^{1+m} \, dx}{1+m}\\ &=-\frac{b x^{2+m}}{2+3 m+m^2}+\frac{x^{1+m} \coth ^{-1}(\tanh (a+b x))}{1+m}\\ \end{align*}

Mathematica [A]  time = 0.0591708, size = 34, normalized size = 0.92 \[ x^m \left (\frac{x \left (\coth ^{-1}(\tanh (a+b x))-b x\right )}{m+1}+\frac{b x^2}{m+2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*ArcCoth[Tanh[a + b*x]],x]

[Out]

x^m*((b*x^2)/(2 + m) + (x*(-(b*x) + ArcCoth[Tanh[a + b*x]]))/(1 + m))

________________________________________________________________________________________

Maple [C]  time = 0.164, size = 676, normalized size = 18.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*arccoth(tanh(b*x+a)),x)

[Out]

1/(1+m)*x*x^m*ln(exp(b*x+a))-1/4*x*(4*b*x+2*I*Pi*csgn(I*exp(2*b*x+2*a))^3+I*Pi*csgn(I*exp(b*x+a))^2*csgn(I*exp
(2*b*x+2*a))*m-I*Pi*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2*m+I*Pi*csgn(I*exp(2*b*x
+2*a))^3*m+I*Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))*m+
4*I*Pi+2*I*Pi*m-4*I*Pi*csgn(I/(exp(2*b*x+2*a)+1))^2-2*I*Pi*csgn(I/(exp(2*b*x+2*a)+1))^2*m-4*I*Pi*csgn(I*exp(b*
x+a))*csgn(I*exp(2*b*x+2*a))^2+I*Pi*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^3*m-2*I*Pi*csgn(I*exp(2*b*x+2*a)
)*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+4*I*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3+2*I*Pi*csgn(I*exp(2*b*x+2*a)
/(exp(2*b*x+2*a)+1))^3-2*I*Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2+2*I*Pi*cs
gn(I*exp(b*x+a))^2*csgn(I*exp(2*b*x+2*a))+2*I*Pi*csgn(I/(exp(2*b*x+2*a)+1))*csgn(I*exp(2*b*x+2*a))*csgn(I*exp(
2*b*x+2*a)/(exp(2*b*x+2*a)+1))-2*I*Pi*csgn(I*exp(b*x+a))*csgn(I*exp(2*b*x+2*a))^2*m-I*Pi*csgn(I/(exp(2*b*x+2*a
)+1))*csgn(I*exp(2*b*x+2*a)/(exp(2*b*x+2*a)+1))^2*m+2*I*Pi*csgn(I/(exp(2*b*x+2*a)+1))^3*m)/(1+m)/(2+m)*x^m

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arccoth(tanh(b*x+a)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.63962, size = 72, normalized size = 1.95 \begin{align*} \frac{{\left ({\left (b m + b\right )} x^{2} +{\left (a m + 2 \, a\right )} x\right )} x^{m}}{m^{2} + 3 \, m + 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arccoth(tanh(b*x+a)),x, algorithm="fricas")

[Out]

((b*m + b)*x^2 + (a*m + 2*a)*x)*x^m/(m^2 + 3*m + 2)

________________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*acoth(tanh(b*x+a)),x)

[Out]

Exception raised: TypeError

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*arccoth(tanh(b*x+a)),x, algorithm="giac")

[Out]

integrate(x^m*arccoth(tanh(b*x + a)), x)